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Beliebt Rechnen >

integral of (6x-5)/(sqrt(x^2-10x+9))

Lösung

\int \frac{6 x - 5}{x ^{2} - 10 x + 9} d x

Lösung

6 x^{2} - 10 x + 9 + 25 ln (\frac{1}{4} x^{2} - 10 x + 9 + x - 5) + C

Schritte zur Lösung

\int \frac{6 x - 5}{x ^{2} - 10 x + 9} d x

Vervollständige das Quadrat

x^{2} - 10 x + 9 : (x - 5)^{2} - 16

= \int \frac{6 x - 5}{( x - 5 ) ^{2} - 16} d x

Wende U-Substitution an

= \int \frac{6 u + 25}{u ^{2} - 16} d u

Multipliziere aus

\frac{6 u + 25}{u ^{2} - 16} : \frac{6 u}{u ^{2} - 16} + \frac{25}{u ^{2} - 16}

= \int \frac{6 u}{u ^{2} - 16} + \frac{25}{u ^{2} - 16} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{6 u}{u ^{2} - 16} d u + \int \frac{25}{u ^{2} - 16} d u

\int \frac{6 u}{u ^{2} - 16} d u = 6 u^{2} - 16

\int \frac{25}{u ^{2} - 16} d u = 25 ln (\frac{1}{4} u^{2} - 16 + u)

= 6 u^{2} - 16 + 25 ln (\frac{1}{4} u^{2} - 16 + u)

Setze in

u = x - 5

ein

= 6 (x - 5)^{2} - 16 + 25 ln (\frac{1}{4} (x - 5)^{2} - 16 + x - 5)

Vereinfache

6 (x - 5)^{2} - 16 + 25 ln (\frac{1}{4} (x - 5)^{2} - 16 + x - 5) : 6 x^{2} - 10 x + 9 + 25 ln (\frac{1}{4} x^{2} - 10 x + 9 + x - 5)

= 6 x^{2} - 10 x + 9 + 25 ln (\frac{1}{4} x^{2} - 10 x + 9 + x - 5)

Füge eine Konstante zur Lösung hinzu

= 6 x^{2} - 10 x + 9 + 25 ln (\frac{1}{4} x^{2} - 10 x + 9 + x - 5) + C

Graph

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